Besides using our toy model potentials discussed above, we mention another two examples in the following.
In the first example, we adopt the Skyrme-type single-nucleon potential to order \(|\mathbf {k}|^6k_{\mathrm {F}}^3\)  and \(k_{\mathrm {F}}^9\)  from Ref. [1]}. In such a model, the analytical expressions for the \(U_0\)  and \(U_{\mathrm {sym}}\)  are given and the relevant parameters were fitted to fulfill similar empirical constraints as we discussed earlier. The potential-related part of the linear-\(\epsilon \)  correction in () can thus be evaluated straightforwardly to be about \(0.4\,\mathrm {MeV}\)  for the SP6s (SP6m or SP6h) set [1]}. Consequently, the linear-\(\epsilon \)  correction to the total symmetry energy is about \(2.2\epsilon \,\mathrm {MeV}\)  which is close to our toy model calculation. In the second example, we adopt an isospin- and momentum-dependent interaction (MDI) potential that has been used extensively to explore heavy-ion reaction dynamics within transport models [3]}, [4]}. With the MDI potential, the potential-related part of the linear-\(\epsilon \)  correction in () is found to be about \(-2.9\epsilon \,\mathrm {MeV}\) , leading to a total correction of about \(-1.1\epsilon \,\mathrm {MeV}\) , i.e., the 2D symmetry energy would be enhanced.
Furthermore, for a comparison the prediction from using an improve MDI (ImMDI) potential [5]} can also be obtained similarly. In the ImMDI potential model, the momentum-dependence of \(U_0\)  is improved to better fit the experimentally constrained high energy behavior of the nucleon optical potential.
The \(y\)  parameter (in unit of MeV) characterizes the momentum dependence of the \(U_{\mathrm {sym}}\) , i.e., \(\partial U_{\mathrm {sym}}/\partial |\mathbf {k}|>0\)  if \(y>0\) . Tab. REF  gives a summary of the linear-\(\epsilon \)  correction to the symmetry energy from different models investigated in this work.
We notice that the sign of the total correction using the ImMDI potential is not deterministic. For comparisons, in section  we shall extend the perturbative calculations to order \(\epsilon ^2\)  for the symmetry energy and investigate the corresponding effects in 2D, see Tab. REF .
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